[en] During the past years, significant progress has been made in understanding the beam transverse emittance blow-up and its preservation. However, one often finds him-/herself ignorant when he/she tries to explain what was observed in an existing machine or to predict what will happen in a machine under design. There are a number of such examples given in this report. Some of them are even fundamental. These are the challenges. But they are also the directions leading to new achievements. The workshop gladly acknowledged them and promised to work on them